I always try to remember to use "cannot" when applicable as in "I cannot take an umbrella" (because I do not have one).

I also thought that the problem with "I can not take an umbrella" was that it was ambiguous: it could also mean that I could decide not to take an umbrella. Trying to write unambiguously most of the time, I would therefore never use it.

Now I see it claimed that it is in fact not ambiguous, and only carries the second meaning. Usually I would accept such advice at face value and start using "can not" when the second meaning is unambiguously meant, but this comes from a style guide I have other reasons not to respect very much.

I have never encountered the sentence “I can not take an umbrella.” as a whole sentence to mean “I can choose not to take an umbrella.” Does this usage really exist? I know that we say “I can not only take an umbrella but also take a raincoat,” but that is a different situation. I have seen “can not” meaning “cannot,” but I presumed that it was a typo.
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Tsuyoshi ItoNov 3 '10 at 0:43

These two spellings [cannot and can
not] are largely interchangeable, but
by far the more common is “cannot” and
you should probably use it except when
you want to be emphatic: “No, you can
not wash the dog in the Maytag.”